On a few of my packages lately, I have been getting some error 1603 failures. I have tried to do some inquiring via logs, etc, but this error seems a bit vague.

Packages I have tried pushing that have resulted sporadically in a 1603 error include:

Civil 3D 2012
AutoTurn 8

There does not seem to be a pattern or specific O/S, etc. that the error is confined to, which would lead me to believe that there is something wrong with the particular machine perhaps or a permissions issue not set correctly?

1603 errors occur for various reasons some of which include, access denied for a some resource which the installation is trying to access say a folder, a network drive or mapped drive location, etc. As suggested by andys0123 try analyzing the log file and you must be able to figure out the issue.
You could post the log file here in case you have so that we might be able to look at it as well.